I have Send as permission on a mailbox. When I log on to my mailbox and send an item as the other user, why does the item appear in my Sent Items folder, not the other user's folder?

This behavior is a feature, not a bug. When you send an item, you're the person who's sending it, even if you're doing so on behalf of another user. This behavior is analogous to the way real-world systems work: If I send a fax from my fax machine, the recipient sees my fax number, even if the cover sheet says that the fax is from the president of the United States.